This paper gives a shortest path algorithm for CFG (context free grammar) labeled and
weighted digraphs where edge weights may be positive or negative,
but negative-weight cycles are not allowed in the underlying unlabeled graph.
These results build directly on an algorithm of Barrett et al. [SIAM J. Comput.30 (2000) 809–837].
In addition to many other results, they gave a shortest path algorithm for CFG labeled and
weighted digraphs where all edges are nonnegative.
Our algorithm is based closely on Barrett et al.'s algorithm as well as Johnson's algorithm
for shortest paths in digraphs whose edges may have positive or negative weights.

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